Streamline diffusion - Alchetron, The Free Social Encyclopedia

Streamline diffusion, given an advection-diffusion equation, refers to all diffusion going on along the advection direction.

Explanation

If we take an advection equation, for simplicity of writing we have assumed ∇ ⋅ u = 0 , and | | u | | = 1

∂ ψ ∂ t + u ⋅ ∇ ψ = 0.

we may add a diffusion term, again for simplicty, we assume the diffusion to be constant over the entire field.

D ∇ 2 ψ ,

Giving us an equation of the form:

∂ ψ ∂ t + u ⋅ ∇ ψ + D ∇ 2 ψ = 0

We may now rewrite the equation on the following form:

∂ ψ ∂ t + u ⋅ ∇ ψ + u ( u ⋅ D ∇ 2 ψ ) + ( D ∇ 2 ψ − u ( u ⋅ D ∇ 2 ψ ) ) = 0

The term below is called streamline diffusion.

u ( u ⋅ D ∇ 2 ψ )

Any diffusion orthogonal to the streamline diffusion is called crosswind diffusion, for us this becomes the term:

( D ∇ 2 ψ − u ( u ⋅ D ∇ 2 ψ ) )

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